Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=20-n $$

Step-by-Step Solution

Answer

The first five terms of the sequence are 19, 18, 17, 16, and 15.

Step 1: Understand the Sequence Rule

The sequence is defined by the rule \(a_n = 20 - n\). This means that the nth term of the sequence is found by substituting \(n\) into the expression \(20 - n\).

Step 2: Calculate the First Term

To find the first term \(a_1\), substitute \(n = 1\) into the expression: \[ a_1 = 20 - 1 = 19 \]

Step 3: Calculate the Second Term

To find the second term \(a_2\), substitute \(n = 2\) into the expression: \[ a_2 = 20 - 2 = 18 \]

Step 4: Calculate the Third Term

To find the third term \(a_3\), substitute \(n = 3\) into the expression: \[ a_3 = 20 - 3 = 17 \]

Step 5: Calculate the Fourth Term

To find the fourth term \(a_4\), substitute \(n = 4\) into the expression: \[ a_4 = 20 - 4 = 16 \]

Step 6: Calculate the Fifth Term

To find the fifth term \(a_5\), substitute \(n = 5\) into the expression: \[ a_5 = 20 - 5 = 15 \]

Key Concepts

Understanding Algebra in Sequences

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In the context of sequences, algebra helps us define and understand the rule that generates the terms of a sequence. The rule is presented as a formula, like the one in our exercise: \[ a_n = 20 - n \]. This expression uses algebra to show how any term in the sequence is derived using the term's position, denoted by \(n\). The algebraic formula allows us to swap different values of \(n\) to find specific terms within the sequence. Essentially, algebra provides the skeleton for systematically working through sequences and finding each term. This makes algebra an invaluable tool for anyone looking to understand sequences quickly and effectively.

Calculating the First Five Terms

Finding the first five terms of a sequence gives us an initial glimpse into its behavior. Here, our sequence rule is \(a_n = 20 - n\). By plugging consecutive integer values of \(n\) from 1 to 5 into this rule, we can calculate the first five terms.

• First Term: Substitute \(n=1\) into the formula to get: \[ a_1 = 20 - 1 = 19 \]
• Second Term: For \(n=2\), it's \[ a_2 = 20 - 2 = 18 \]
• Third Term: For \(n=3\), it's: \[ a_3 = 20 - 3 = 17 \]
• Fourth Term: For \(n=4\), it's: \[ a_4 = 20 - 4 = 16 \]
• Fifth Term: For \(n=5\), it's: \[ a_5 = 20 - 5 = 15 \]

Calculating these terms helps solidify our understanding of how each subsequent term decreases by 1, as determined by the given formula.

The Role of the Sequence Formula

The sequence formula is a mathematical expression that describes the relationship between the terms of a sequence and their positions. In this case, the formula \(a_n = 20 - n\) is crucial because it not only defines the sequence but also dictates its pattern. This pattern is recognized through the consistent change applied as \(n\), the position, increases.The sequence formula enables:

• Prediction: You can anticipate the value of any term without listing all prior terms.
• Pattern Analysis: Understand how each term relates to others and to the sequence as a whole.
• Flexibility: Easily change the position \(n\) to explore any term you need.

Without such a formula, finding terms would be cumbersome and prone to error, especially in more complicated sequences.

Understanding the nth Term in a Sequence

The "nth term" of a sequence is a way to express a position within a sequence using a variable, usually denoted as \(n\). For the given sequence rule \(a_n = 20 - n\), knowing the value of \(n\) allows us to compute the corresponding term directly.The concept of the nth term is powerful:

• It provides a general way to determine any term in the sequence.
• It makes it possible to look forward and predict future terms beyond a limited dataset.
• It serves as a quick check for verifying patterns within the sequence.

In our exercise, understanding the nth term lets us see a clear pattern, revealing that as \(n\) increases, the term value decreases linearly by 1. This understanding is key for both practical problem-solving and laying the foundation for more advanced mathematical topics.